Anybody with exposure to music-theory education has seen a circle of fifths similar to the one at the end of this post. A handy diagram showing the twelve key signatures neatly sorted by number of accidentals, sharps to the right, flats on the left.

Musicians use the circle as a compass to navigate the major and minor key signatures. But where does the circle of fifths come from? Why do keys a perfect fifths apart always differ by exactly one accidental? And are there other uses for circle-of-fifths?

Answering two even more basic questions explains all of the above and touches on some of the most basic principles of music in general and western music in particular:

- Why a circle?
- Why fifths?

## Why a Circle?

One of the simplest diagrams places pitches from lowest to highest along a straight line.

In this diagram neighboring notes are half-steps apart. They are separated by the smallest difference in pitch available in a twelve tone system. Playing a section of the notes on the line produces a Chromatic Scale.

This “Line of Pitches” is so fundamental because it links the musical system of pitches (the notes we can play on musical instruments and their names) to a physical property: Frequency. In the line of pitches frequency increases from left to right.

A linear order of pitches is inherent to many musical instruments. Sometimes by the instrument designer’s choice (e.g. keyboard instruments) or as a result of the physics of an instrument’s tone production mechanism. For example string instruments where moving up the neck and shortening a string produces higher pitches.

### Octave Equivalence

Often in musical practice it is not that important which octave a note is from. We perceive pitches an octave apart to have the same fundamental quality. This principle is called Octave Equivalence.

Octave equivalence is one of the few musical principles shared across all cultures (Jourdain). It is the reason why after one octave the names of the notes repeat even though the notes are not identical (one being an octave higher than the other).

Using octave equivalence, we could simplify the line of pitches diagram to only span one octave. But such a diagram wouldn’t tell us that F#’s upper neighbor is G. One way of fixing that problem is to reconnect F# and G by “bending” the line into a circular shape and closing the gap. Such a diagram is called the Chromatic Circle.

### Visualizing Intervals and Chords

The chromatic circle is a great tool for visualizing and thinking about intervals.

Take for example the C diminished 7th chord (C°7) consisting of three stacked minor thirds. On the chromatic circle, moving up a minor third corresponds to moving 90° clockwise. The following diagram shows C°7 on the chromatic circle. The notes from the chord are drawn as red dots.

The symmetrical nature of the diminished 7th chord is clearly reflected in the diagram. It is also easy to see, that there are only three different diminished 7th chords possible: We can only rotate the “square-of-minor-thirds” two times by a half-step before we hit the same shape again.

## Why Fifths?

Of the 11 intervals in a twelve-tone system, only four enumerate all twelve notes when applied repeatedly:

- The
**Minor Second**(i.e. one half step). The interval used to create the chromatic cycle. Starting on C and moving up in minor seconds, we get the sequence: C, C#, D, D#, E, F, F#, G, G#, A, A#, B: The notes on the chromatic cycle. - The
**Major Seventh**(eleven half steps). This is the inverse interval of the minor second. By repeatedly moving up in major sevenths, we get the sequence: C, B, Bb, A, Ab, G, Gb, F, E, Eb, D, Db. These are the notes of the chromatic cycle counter-clockwise, i.e. descending chromatically. - The
**Perfect Fifth**(seven half steps). By repeatedly moving up in perfect fifths, starting at C we get the sequence: C, G, D, A, E, B, F#/Gb, Db, Ab, Eb, Bb, F. The notes on the Circle-of-Fifths. - The
**Perfect Fourth**(five half steps). The inverse interval of the perfect fifth. It produces the sequence C, F, Bb, Eb, Ab, Db, Gb/F#, B, E, A, D, G: the same as going around the circle-of-fifths counter-clockwise.

Any other interval applied repeatedly returns to its starting note before it reaches all twelve notes. The diminished chord from the previous section is an example of an interval (the minor third) returning to its starting point after only 4 steps.

In addition to enumerating all twelve notes, perfect fifths are also the most harmonious interval after the octave. Therefor adjacent notes on the circle of fifths are harmoniously most closely related.

Contrast that with the chromatic circle, where adjacent notes are closest in terms of pitch or frequency, but harmoniously they are related by one of the most dissonant intervals, the minor second!

### Diatonic Scales

The system of western music is mostly based on the diatonic scales (i.e. major scales and their modes). Diatonic scales are based on the perfect fifth. The circle of fifths is particularly useful for analyzing and organizing diatonic scales.

The fact that diatonic scales are constructed from perfect fifths can easily be seen by highlighting the notes of the C-Major scale on a circle of fifths similarly to what we did with the diminished chord and the chromatic circle previously.

The notes of the major scale cover a contiguous section of the circle! The endpoints of that section are on opposite sides of the circle. The interval between diametrically opposed notes is the Tritone (also called a diminished fifth or augmented fourth).

Given a circle of fifths, one can easily find the notes making up any major scale using the following recipe:

- Find the scale’s root on the circle and mark it. Example: If the goal is to construct an Eb-Major scale, this would be Eb at the 9 o’clock position.
- Mark the counter-clockwise neighbor of the root note, i.e. the note “one hour before the root”. For Eb that would be Ab at the 8 o’clock position.
- Mark the next five notes clockwise from the root. In the Eb example those would be Bb, F, C, G and D. Or in other words the range from 10 o’clock to 2 o’clock.

The circle of fifths is commonly used to organize the twelve major keys and their key signatures. Neighboring keys in the circle share all but one note. Because all the notes in a major scale cover a contiguous section of the circle one can simply rotate that section to the left or the right to get the notes of the neighboring keys.

For example, when going from the key of C (see the image above) to the key of G, one would rotate the half-circle covering the section from F the B over “one hour”, making it start at C and end at F#.

Another way of thinking of this rotation is to move the F from the left end of the section over to the other side of the circle where it becomes an F# effectively raising it by a half step. That is the reason why the key signature of G-Major contains one sharp, the F#.

When going through keys around the circle of fifths, a sharp is added each time one moves up a fifths (clock-wise). The opposite happens, when moving counter clockwise. With each key a fifth lower one note needs to be lowered by a half step. That is the reason why the number of flats in the key signatures increases when moving through the circle of fifths counter clockwise.

The following is a typical circle including the major keys, their key signatures and their relative minor keys.

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